Linear Algebra and SVD(Some slides adapted from Octavia Camps)
Represent points as column vectors.
Represent motion as matrices.
Move geometric objects with matrix
2D Translation Equation
2D Translation using Matrices
Counter-clockwise rotation by an angle Counter-clockwise rotation by an angle
Why does multiplying points by R rotate them?
Think of the rows of R as a new coordinate system.
Taking inner products of each points with these expresses
that point in that coordinate system.
This means rows of R must be orthonormal vectors
(orthogonal unit vectors).
Think of what happens to the points (1,0) and (0,1). They
go to (cos theta, -sin theta), and (sin theta, cos theta).
They remain orthonormal, and rotate clockwise by theta.
Any other point, (a,b) can be thought of as a(1,0) +
b(0,1). R(a(1,0)+b(0,1) = Ra(1,0) + Ra(0,1) = aR(1,0)
+ bR(0,1). So its in the same position relative to the
rotated coordinates that it was in before rotation
relative to the x, y coordinates. That is, its rotated.
Degrees of Freedom
R is 2x2 R is 2x2 4 elements4 elements
BUT! There is only 1 degree of freedom: BUT! There is only 1 degree of freedom:
The 4 elements must satisfy the following constraints:The 4 elements must satisfy the following constraints:
Transformations can be
Matrix multiplication is associative.
Combine series of transformations into
one matrix. (example, whiteboard).
In general, the order matters.
2D Rotations can be interchanged.
Rotation and Translation
tycos qsin q
tx-sin qcos q( )(1
)Rotation, Scaling and Translation
Rotation about an arbitrary point
Can translate to origin, rotate, translate
back. (example, whiteboard).
This is also rotation with one translation.
Intuitively, amount of rotation is same
But a translation is added.
Stretching = tilting and projecting
(with weak perspective)
For any matrix, M = R1DR2 R1, R2 are rotation matrices
D is a diagonal matrix.
This decomposition is unique.
Efficient algorithms can compute this (in
Simple 3D Rotation
Rotation about z axis.
Rotates x,y coordinates. Leaves z coordinates fixed.
Full 3D Rotation
Any rotation can be expressed as combination of three
rotations about three axes.
Rows (and columns) of R are
R has determinant 1 (not -1).
Intuitively, it makes sense that 3D rotations can be
expressed as 3 separate rotations about fixed axes.
Rotations have 3 degrees of freedom; two describe an
axis of rotation, and one the amount.
Rotations preserve the length of a vector, and the
angle between two vectors. Therefore, (1,0,0), (0,1,0),
(0,0,1) must be orthonormal after rotation. After rotation,
they are the three columns of R. So these columns
must be orthonormal vectors for R to be a rotation.
Similarly, if they are orthonormal vectors (with
determinant 1) R will have the effect of rotating (1,0,0),
(0,1,0), (0,0,1). Same reasoning as 2D tells us all other
points rotate too.
Note if R has determinant -1, then R is a rotation
plus a reflection.
3D Rotation + Translation
Just like 2D case